Causal Theory in Power Space

George Ellis and a few other physicists have recently introduced the idea of top-down causation in fundamental physics. In the context of classical spacetime, the hypothesis of top-down causation is that causal relationships among subsets of spacetime are not completely reducible to causal relations among their constituent events.

Causal set theory, by contrast, is purely bottom-up at the classical level. In causal set theory, causality is modeled as an irreflexive, acyclic, locally finite binary relation on a set, whose elements are viewed as spacetime events. Since causal structure alone is not sufficient to recover metric structure in the continuum limit, measure-theoretic data is also supplied in the form of a discrete constant measure. Rafael Sorkin, the founder of causal set theory, summarizes these two ingredients with the phrase “order plus number equals geometry.” This statement is a special case of what I call the causal metric hypothesis.

If the hypothesis of top-down causation is true, then causal structure exists not at the level of spacetime itself, but at the level of the power set of spacetime, i.e., the set whose elements are subsets of spacetime. Causal graphs may still be employed to describe such a structure, but the vertices now correspond to families of events, rather than only single events. A possible concern is violation of locality, but this is not an insurmountable problem; it merely requires appropriate constraints among high-level and low-level relations. I call the resulting theory causal theory in power space.

I will say a few words about why this idea is physically attractive. First, the arguments in favor of top-down causation are rather convincing in their own right. Second, one of the principal difficulties for pure causal theory is its parsimony; it is not clear that it contains enough structure to recover established physics. Top-down causation, employed as I described above in terms of power-set relations, provides “extra structure without extra hypotheses.” The types of mathematical objects and the general techniques are unchanged; it is only their interpretation that is generalized. In particular, the causal metric hypothesis still applies, although not in the form “order plus number equals geometry.”

There is considerable precedent, at least in mathematics, for this type of generalization. For example, Grothendieck’s approach to algebraic geometry involves “higher-dimensional points” corresponding to subvarieties of algebraic varieties, and the explicit consideration of these points leads to the theory of algebraic schemes, which has considerable advantages. In particular, the scheme structure on a variety is consistent with the variety structure, but contains useful additional information. Similarly, higher-level causal structure, suitably constrained, is consistent with lower-level causal structure, but contains important additional information that might be essential in recovering established physics. The holistic aspects of causal theory in power space might aid in establishing the shape/causal duality discussed in my previous post.